Mellor, who has pursued a high media profile since resigning from the Government last September, takes over from the Greek shipping millionaire and philanthropist, Eddie Kulukundis, who held the position for five years.
During Mellor's first year as chairman, the SAF - an independent organisation established in 1976 to help fund British sportsmen and women competing in major international events - will distribute pounds 1.4m in grants.
Last year's recipients included the Olympic 800 metres finalist, Curtis Robb, and the skier Emma Carrick-Anderson, who was eighth in the combined slalom at last year's Winter Olympics.
Mellor made it clear that he envisages raising the level of support for Britain's young sportsmen and women. 'We expect our athletes to be world beaters when firstly they have got to beat the system before they even get a chance,' he said.
'When Spain knew they were going to host the 1992 Olympics, extra resources were made available for training their athletes, and what do you know - they got 13 gold medals. That proves the point that talent is not enough. It needs a framework.'
This year's SAF grants have been made up with around pounds 250,000 from their own lottery scheme, which Mellor foresees continuing alongside the National Lottery, and a pounds 786,000 donation from the Foundation for Sports and the Arts. But the latter award was limited to non-elite sportsmen and women, and Mellor expressed frustration at the currrent maximum SAF grant of pounds 6,000 to any individual.
'Surely to heaven we have got to unleash some more of our resources,' he said.
'Let's put our back into British sport and then in four years' or eight years' time we can really give them a show.'